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Solid State

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Solid State ( Crystal Vibration - Phonos)
Qn 1.
Find an expression for the frequency of the elastic waves in terms of wave vector k.
A wave vector refers to a vector which describes a wave. Like any other vector, it comes with
both magnitude and direction,
The magnitude could be the wavenumber or the angular wavenumber which is inversely proportional
to wavelength and the direction is the direction of the wave propagation
For a perfect one-dimensional wave, the equation is;
ψ(x, t) = A cos( kx ωt + φ)
where
x = position,
t = time,
ψ is a function of x and t disturbance of the wave (e.g for water wave, ψ is the height of water,
and in sound wave, it would be air pressure).
A = amplitude of wave (peak oscillation),
φ = "phase offset" (shows how the waves sync together)
ω = temporal angular frequency,( shows the number of oscillations completed in a unit time,
relating to the periodic time T in the equation ω=2∏/T,
K = spatial angular frequency of the wave, (shows number of oscillations completed in a unit
relating to wavelength by equation K = 2π/ג.
The wave moves towards the direction +x with speed (phase velocity) . ω/K

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Qn 2.
Find the range of k physically significant for elastic waves in crystal from the condition the slope of
2
vsk vanishes at Brillouin zone boundary.
First Brilouin zone
What range of k is physically significant for elastic waves?
Only those in the first Brilloiun.


= e
ika
The range π to π for the phase ka covers all independent values of the exponential.
Qn 3.
Show that the group velocity, that is velocity of energy propagation in the medium v
g
=dm/dk
In terms of the wave vector k times a (the spacing between planes)
The group velocity in waves is the velocity which the overall shape of the waves' amplitudes
propagates through space
Consider a wave packet as a function of position x and time t: α(x,t).
Let A(k) be its Fourier transform at time t=0,
ɑ(x, 0) =

e
ikx

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Solid State ( Crystal Vibration - Phonos) Qn 1. Find an expression for the frequency of the elastic waves in terms of wave vector k. A wave vector refers to a vector which describes a wave. Like any other vector, it comes with both magnitude and direction, The magnitude could be the wavenumber or the angular wavenumber which is inversely proportional to wavelength and the direction is the direction of the wave propagation For a perfect one-dimensional wave, the equation is; ψ(x, t) = A cos( kx – ωt + φ) where • x = position, • t = time, • ψ is a function of x and t disturbance of the wave (e.g for water wave, ψ is the height of water, and in sound wave, it would be air pressure). • A = amplitude of wave (peak oscillation), • φ = "phase offset" (shows how the waves sync together) • ω = temporal angular frequency,( shows the number of oscillations completed in a unit time, relating to the periodic time T in the equation ω=2∏/T, • K = spatial angular frequency of the wave, (shows number of oscillations completed in a unit relating to wavelength by equation K = 2π/‫ג‬. The wave moves towards the direction +x with speed (phase velocity) . ω/K Qn 2. Find the range of k physically significant for elastic waves in crystal from the condition the slope of ∞2 vsk vanishes at Brillouin zone boundary. First Brilouin zone What range of k is physically significant for elastic waves? ⇒ Only those in the first Brilloiun. 𝑈 (𝑠+1) ? ...
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